238, 239] INDUCTION OF CURRENTS. 485 



which gives rise to the difference of potential, or electrostatic 

 electromotive-force impressed in the circuit in the direction of the 

 current, 



*----* 



Accordingly the differential equation for the current is 



from which, substituting from (i), we obtain the equation for the 

 charge, 



Again, assuming q = e xt we obtain the quadratic for X 



(6) 



whose roots are 



R 



~ 



Z 2 KL' 



We have now to consider two cases. 



CASE I. R* > 4tL/K. Both roots real We then have 



(8) q = Ae^J 4- Be* zt , 



and as \ and X 2 are both negative, the charge, and likewise the 

 current 



(9) / = \iAe* it + Afclfe***, 



die gradually away. If there is a permanent impressed electro- 

 motive-force E in the circuit, we must add the quantity E K to 

 the charge, which, however, does not affect the current. 



Determining the constants A and B by the conditions that 

 there is initially neither current nor impressed electromotive- 

 force, and that the initial charge is q Q , we have 



Oo) 1 = ^ 



while if there is no initial charge, but an impressed electromotive- 

 force E 0j we obtain 







